On Friday in 121, we continued our discussion of ways to find absolute and local extreme points. Our biggest problem was that we had lots and lots of points to sift through. But we found a trend on Monday that local maxima and minima turned up where the graph of had a "turnaround" of some kind, either a smooth one or a sharp turn. The smooth ones, where are called stationary points, and the sharp turns, or "cusps," occur at singular points, where doesn't exist. Together these two types of points make up the critical points for f. The critical points are the candidates for locations of local maxima and minima. In fact, if a function has a local maximum or minimum at c, then c must have been a critical point. (Note that all critical points must be in the domain of f.)

We then looked at a couple of examples. We saw that in the case where can be written as a fraction, the critical point question becomes one of when the numerator is 0, and when the denominator is 0.

We then applied this technique to the Extreme Value Theorem and came up with a procedure to find the absolute maxima and minima of our function f on a closed interval [a,b]:

1. Find the values of f at the critical points inside the interval [a,b].
2. Find the values of f at a and b.
3. The largest of the above values is the absolute maximum value on [a,b]; the smallest is the absolute minimum value.

We'll discover next week how to classify critical points as maxima, minima, or none of the above.